Rudolf Peierls’ 1939 Analysis of Critical Conditions in Neutron Multiplication

By B. Cameron Reed

December 2008 marks the 70th anniversary of the discovery of fission by Hahn and Strassmann, one of the most pivotal scientific discoveries of the twentieth century. The story of the subsequent elucidation of the fission process and the development and use of nuclear weapons continues to hold a strong fascination for physicists, historians, and laypersons alike. In this article I briefly examine a side story to this history that is now often overlooked: the first detailed analysis of the physics involved in estimating critical mass, an analysis published by Rudolf Peierls in late 1939.

Peierls’ paper [1] was received by the Proceedings of the Cambridge Philosophical Society on June 14, 1939 and published in October of that year. The now relative obscurity of this paper is likely attributable to a combination of reasons: fission had been observed only with slow neutrons at the time and so any prospect of a weapon must have seemed remote if not impossible, it was published just after the appearance of Bohr & Wheeler’s extensive analysis of fission in the September 1, 1939 Physical Review, and the fact that Peierls did not apply the formulae he developed to any situation as he lacked reliable estimates for cross-sections and secondary neutron numbers. By July 1941 the British MAUD report [2] used his formulae to estimate a critical mass for U-235 of about 9 kg. While this is an underestimate compared to the currently-accepted value of ~ 45 kg (a consequence of optimistic parameters; see [3]) one cannot help but wonder if he would have published in the open literature had he a sense of the numbers in the summer of 1939.

Discussions of the technicalities of computing critical mass now typically refer to the diffusion-theory approach presented in Robert Serber’s Los Alamos Primer [4]. Peierls’ name did not appear in the original Primer but does in Serber’s book by virtue of the fact that it includes a reprint of the Frisch-Peierls memorandum of March 1940. It is of interest, then, to examine how the predictions of Peierls’ formulae compare to those of diffusion theory upon adopting modern values for the fission parameters.

Peierls parameterized fissility with a dimensionless variable he designated as ξ and defined as,


where σf and σs are the fission and scattering cross-sections and where ν is the number of secondary neutrons emitted per fission; non-fission neutron absorption is ignored here. Note that if the neutron multiplicity is low (ν ~ 1) then ξ → 0 whereas if ν >> 1 then ξ → 1, that is, 0 ≤ ξ ≤ 1 in general. He then developed formulae for the critical radius R of the form (βR)-1= f(ξ) that apply in these two extremes, where  β = ns + νσf) and where n is the nuclear number density. Explicitly, f(ξ) is given by

As nature would have it, the situation in reality for U-235 falls into neither of these extremes but rather takes ν ~ 2.6 and ξ ~ 0.51.

The solution to the diffusion equation for the critical radius can only be carried out numerically as described in references [3] and [4]. In terms of a reduced critical radius x = R/d where

the diffusion equation leads to the condition



Here λf and λt are respectively the neutron mean free paths for fission and transport, defined as (λf)-1 = nσf and  (λf)-1 = nf  + σs); σs is here taken to be the cross-section for elastic scattering only, as effects of inelastic scattering are ignored in Serber’s treatment.

It is straightforward to show that in terms of Peierls’ (βR)-1 quantity, the solution of Eq. (4) corresponds to


The accompanying graph shows runs of (βR)-1 (in meter-3) for the physically interesting range of ξ. The dashed curves are Peierls’ expressions for ξ → 1 and ξ → 0; the solid line is the diffusion theory prediction. Peierls’ curves actually converge at ξ → 1 although only one of them is valid there. Notice that the diffusion-theory curve tracks closely to Peierls’ ξ → 0 curve; this is because the diffusion theory is in fact only strictly valid when the size of the bomb core is large in  comparison with the neutron mean free paths involved, which is the case when ν ~ 1. (Even for relatively large values of ξ, however, the diffusion-theory prediction tracks reasonably closely to Peierls’ ξ → 1 curve.) As explained by Serber, a more exact treatment (which he does not detail) gives results in close accord with those of diffusion theory.

Adopting average fission parameters for U-235 as given in reference [3], (σf, σs, ν) = (1.235 bn, 4.566 bn, 2.637) gives ξ = 0.5083. Solving the diffusion equation gives βR ~ 3.1378 whereas Peierls’ solutions give ~ 2.9726 and 3.5907 for ξ → 0 and ξ →1. The mean of Peierls’ solutions lies only about 4.6% higher than the diffusion-theory solution; this would correspond to overestimating the critical mass by about 14% in comparison with the diffusion solution. The diffusion solution corresponds to a critical radius of about 8.26 cm, or a critical mass of ~ 45 kg. For Pu-239 (ξ = 0.6221) the various solutions are in even closer accord with the mean of Peierls’ solutions giving a critical radius only about 3.2% lower than that from diffusion theory.

Clearly, Peierls developed an accurate and quite general model for predicting critical masses within a few months of the discovery of fission. But does this imply that fission weapons might have been available earlier had his work in some sense been better appreciated at the time? In the opinion of this author this is not likely: he had no experimental values available for the fissility parameters and apparently did not consider the idea of a fast-fission pure U-235 bomb until approached about it by Otto Frisch in early 1940. Even then they had to base their estimate of critical mass (about one pound) on an estimate of the fission cross-section derived from scattering theory. Experimental uncertainties aside, one has to admire Peierls’ treatment of the problem. The availability of his work in the open literature at the outbreak of World War II makes all the more remarkable Werner Heisenberg’s famous misunderstanding of the issue.


  1. R. Peierls, “Critical Conditions in Neutron Multiplication,” Proceedings of the Cambridge Philosophical Society 35, 610-615 (1939).
  2. The MAUD report can be found in its entirety in M. M. Gowing, Britain and Atomic Energy 1939-1945 (St. Martins Press, London, 1964); see pp. 402-405 for the report’s discussion of fission parameters and estimated critical masses.
  3. B. C. Reed, “Arthur Compton’s 1941 Report on explosive fission of U-235: A look at the physics,” Am. J. Phys. 75 (12), 1065-1072 (2007).
  4. R. Serber, The Los Alamos Primer: The First Lectures on How To Build An Atomic Bomb (Univ. of California, Berkeley, CA, 1992).

B. Cameron Reed
Department of Physics, Alma College
Alma, MI 48801
ph: (989) 463-7266

This contribution has not been peer refereed. It represents solely the view(s) of the author(s) and not necessarily the views of APS.